An Approach to Non-Abelian Cyclotomic Fields

Abstract

We mainly study a polynomial $f_{1,n}(x)=x^{n-1}+2x^{n-2}+3x^{n-3}+\cdots +k{x}^{n-k}+\cdots +(n-1)x+n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root ${\alpha }_n$ of $f_{1,n}(x)$ satisfies $|{\alpha }_n|\to 1$ ($n\to \infty$), and discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ for several type $n$. After that, we show that the Galois group of $f_{1,n}(x)$ is Symmetric group $S_{n-1}$ for several type $n$. Although those roots of $f_{1,n}(x)=0$ don't draw an exact circle, it looks like a circle on complex plane. Moreover by considering that Galois groups of $f_{1,n}(x)$ are not abelian in many cases, we call such extension fields over $\mathbb{Q}$ "Non-Abelian Cycrotomic Fields" here.

15 pages